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This question isn't really very clear or practical. You need to add quite a bit of information, eg which planet?

If Sedna vanished, it would be detectable by instruments, but we would not notice.

Jupiter or Mars might make a bigger difference. We could spot it straight away by the naked eye.

Of course Earth life would be fine at first (unless it was Earth that was removed ...) but the orbits of asteroids and other bodies may be perturbed, which could lead to an influx of asteroid strikes - potentially - in the longer term.

Short answer: the solar system would be less stable in the short run due to the massive gravitational influence of Jupiter vanishing.

Long answer: The current layout of the solar system is actually remarkably well balanced; without outside interference, it'll stay stable for a good long while. Anything that changes the current equilibrium will force the system to adjust until it finds a new equilibrium. Removing a huge mass will certainly affect this in some way, but it's difficult to say exactly how without running a few hundred simulations. Jupiter's gravitational influence kept the asteroid belt from coalescing into a planet, so that might happen, for one.